In physics, topological order describes a state or phase of matter that arises in quantum mechanics. Whereas classical phases of matter such as gases and solids correspond to microscopic patterns in the spatial arrangement of particles, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.
Technically, topological order occurs at zero temperature. Various topologically ordered states have interesting properties, such as (1) ground state degeneracy and fractional statistics or non-abelian group statistics that can be used to realize a topological quantum computer; (2) perfect conducting edge states that may have important device applications; (3) emergent gauge field and Fermi statistics that suggest a quantum information origin of elementary particles; (4) topological entanglement entropy that reveals the entanglement origin of topological order, etc. Topological order is important in the study of several physical systems such as spin liquids, and the quantum Hall effect, along with potential applications to fault-tolerant quantum computation.
Topological insulators and topological superconductors (beyond 1D) do not have topological order as defined above, their entanglements being only short-ranged, but are examples of symmetry-protected topological order.